Alexander and Thurston norms of graph links

Abstract.

We show that the Alexander and Thurston norms are the same for all irreducible Eisenbud-Neumann graph links in homology 3-spheres. These are the links obtained by splicing Seifert links in homology 3-spheres together along tori. By combining this result with previous results, we prove that the two norms coincide for all links in S3 if either of the following two conditions are met; the link is a graph link, so that the JSJ decomposition of its complement in S3 is made up of pieces which are all Seifert-fibered, or the link is alternating and not a (2,n)-torus link, so that the JSJ decomposition of its complement in S3 is made up of pieces which are all hyperbolic. We use the E-N obstructions to fibrations for graph links together with the Thurston cone theorem on link fibrations to deduce that every facet of the reduced Thurston norm unit ball of a graph link is a fibered facet.

Key words and phrases:

2000 Mathematics Subject Classification:

1.1. The Alexander norm (A-norm)

A semi-norm defined on the first cohomology group H1(M;Z) of a connected, compact, orientable 3-manifold M, whose boundary (if any) is a union of tori, was introduced by McMullen [8] in the late 1990s. It is directly determined by the multivariable Alexander polynomial △ of the 3-manifold M and is called the Alexander norm of M. We shall adopt the notation A-norm for the Alexander norm.

If we write a multivariable Alexander polynomial △ in multi-index notation, then △=∑α∈supp(△)cαtα with tα=tα11…tαnn and supp(△) denoting the support of the polynomial △; the set of all α labeling non-zero constants cα. Let α and β be the exponents of any two arbitrarily chosen terms of △. They are elements of H1(M;Z). Let ϕ be a cohomology class in the space dual to the space containing the exponents so that ϕ∈H1(M;Z)=Zb1 where b1 is the first Betti number of M. The A-norm ∥ϕ∥A of ϕ is the supremum of ϕ(α−β) taken over all the exponents α,β in the support of △.

1.2. The Thurston norm (T-norm)

The A-norm is closely related to a semi-norm for compact, oriented 3-manifolds introduced in 1986 in [13], called the Thurston norm. We shall adopt the notation T-norm for the Thurston norm. In the compact, oriented 3-manifold M with a boundary ∂M which may be empty, every homology class z∈H2(M,∂M;Z) can be represented as [S] where S is a properly embedded, oriented surface. By Poincaré duality the relative second homology group is isomorphic to the first cohomology group, H2(M,∂M;Z)≅H1(M,Z). The T-norm ∥ϕ∥T is determined by the Euler characteristic χ of the surface S representing the cohomology class ϕ through this duality isomorphism.

1.3. Eisenbud-Neumann (E-N) graph links

An E-N graph link (see [3, 10]) is a link L=L1∪⋯∪Lr in a homology 3-sphere Σ, which can be built up by splicing together Seifert links in homology 3-spheres. We use the notation L=(Σ,L) to indicate not only the link L but also the homology 3-sphere Σ it lies in. To each graph link L there is a unique minimal graph. It is shown in [3] that the graph directly determines the Alexander polynomial and Thurston norm of L.

1.4. Main theorem: The coincidence of Alexander and Thurston norms for graph links

Theorem 1.5.

The Alexander and Thurston norms coincide for all irreducible graph links with two or more components.

We prove this theorem by directly calculating the A-norm of graph link L from the expression for its Alexander polynomial, and comparing it to the expression for its T-norm. We derive three new results which are used in the proof. Each of these results involves a sum over the p splice components of the graph of L;

a T-norm decomposition formula,

the Newton polyhedron of the Alexander polynomial of L as a Minkowski sum, and

an A-norm decomposition formula.

In the case of a graph knot, which is a graph link with one component, we show as a corollary that the two norms satisfy ∥ϕ∥A−∥ϕ∥T=|ϕ|.

Without actually calculating the two norms themselves, McMullen proved in [8] that for a compact, connected, oriented 3-manifold M whose boundary if any is a union of tori, the two norms must coincide for all fibered cohomology classes which are primitive. A primitive cohomology class ϕ is one whose image is Z so that ϕ(H1(M;Z))=Z. Hence our result that the two norms coincide for graph links is new with regard to non-primitive cohomology classes and the non-fibered cohomology classes.

In [8], McMullen states that the two norms, Alexander and Thurston, coincide for many 3-manifolds but not all. Examples of links for which the two norms do not coincide are given by a two-component link found by Dunfield [2] and the link 9321 (See [8]).

1.6. Generalization of the main theorem

As an application of our main result, we combine it with previous results to obtain a more general theorem in the context of the JSJ (Jaco, Shalen and Johannson) decomposition of a link in S3. According to the JSJ decomposition, the link complement X can be decomposed into irreducible pieces of two types, Seifert-fibered and hyperbolic, by de-splicing the pieces together along tori.

Theorem 1.7.

Let L be a link in S3. Let the JSJ decomposition of the link complement S3∖L consist of k irreducible pieces Ni so that

S3∖L=N1∪TN2∪T⋯∪TNk.

Then the Thurston and Alexander norms of L coincide if either of the following two conditions are met:

L is a graph link so that all the Ni are Seifert-fibered.

L is an alternating link which is not a (2,n)-torus link, so that all the Ni are hyperbolic.

1.8. Characteristic hyperplanes of the Thurston norm ball

T-norm unit ball and fibrations

The T-norm can be extended by convexity from integer-valued classes to real-valued classes to determine a T-norm unit ball. The cohomology classes of the T-norm unit ball can be separated into two types; fibered and non-fibered. An integer-valued cohomology class ϕ is fibered if the surface S which represents it via Poincaré duality is also the fiber F of a fibration of 3-manifold M over the circle S1. A real-valued cohomology class ϕ is fibered if it lies on a ray through the origin whose lattice points are integer-valued fibered cohomology classes. Otherwise ϕ is a non-fibered cohomology class. We do not require the fiber to be connected.

Thurston cone theorem

The Thurston norm unit ball may not be a bounded set. However, we show in Long [6] that by introducing essential coordinates we can define a reduced T-norm unit ball which is a polyhedron of the same dimension as the Newton polyhedron of the Alexander polynomial of 3-manifold M. The Thurston cone theorem (see [13]) states that the set of fibered cohomology classes is some union of the cones pointed at the origin (minus the origin) through the interiors of the facets (top-dimensional faces) of the reduced T-norm unit ball.

Fibration obstruction criterion

Using necessary and sufficient conditions for a cohomology class of a graph link L to be a fibered class from [3], we show that the set of all non-fibered classes of L is made up of a set of hyperplanes we call the characteristic hyperplanes. There is one characteristic hyperplane for each splice component of the graph of L. The ith characteristic hyperplane is made up of those classes whose T-norm for the ith splice component in the Thurston norm decomposition formula is zero.

The fibered facets of the reduced Thurston norm unit ball

We use the fibration obstruction criterion and the Thurston cone theorem to prove the following new theorem on graph link fibrations.

Theorem 1.9.

Every facet F of the reduced T-norm unit ball ~BT for the irreducible graph link L is a fibered facet.

This theorem implies that the boundary of each facet, which is made up of the lower dimensional faces, must be contained in the non-fibered set of cohomology classes.

1.10. Sample calculation

We conclude by applying our results to a sample graph link LEN from Eisenbud and Neumann [3]. We find the Alexander polynomial, Thurston norm, reduced T-norm unit ball and characteristic hyperplanes for this link and determine the intersection of the characteristic hyperplanes with the reduced T-norm unit ball.

2.1. Alexander norm

Let △ be the Alexander polynomial of a compact, connected, orientable 3-manifold M with first Betti number b1, whose boundary (if any) is a union of tori. This polynomial can be expressed using multi-index notation as follows:

△(t1,…,tb1)

=

∑cα1α2…αb1tα11tα22…tαb1b1

=

∑α∈supp(△)cαt% α

where supp(△)={α:cα≠0}. The A-norm is a semi-norm defined on the first cohomology group H1(M;Z) of a 3-manifold M directly determined by the Alexander polynomial △ of M. The Alexander polynomial △ is a Laurent polynomial; △∈Z[t±11,…,t±1b1]. Hence △ can be expressed as a finite sum as in Equation (2.1). Then α∈Zb1,∀α∈supp(△).
Further let ϕ∈(Zb1)∗=HomZ(Zb1,Z)=H1(M;Z)≅Zb1 be an element of the dual vector space (Zb1)∗ to the space Zb1 in which each α lies in. Then there is a semi-norm, which we shall call the A-norm, defined as follows.

Definition 2.2.

The A-norm of ϕ∈H1(M;Z) for the Alexander polynomial △ of M is

(2)

∥ϕ∥A=supα,β∈supp(△)ϕ(α−β).

Remark 2.3.

The definition of the A-norm above can be extended to real-valued cohomology classes ϕ∈H1(M;R) =(Rb1)∗.

Since the A-norm is completely determined by the Alexander polynomial △ we also use the notation ∥ϕ∥A:=∥ϕ∥△. In Long [6], we note that the Alexander norm is the special case of a norm determined by Laurent polynomials in general for which the polynomial is an Alexander polynomial. We call this generalized norm the Laurent norm since each Laurent polynomial f with integer coefficients determines a norm ∥∥f; the Laurent norm for f.

2.4. Thurston norm

The T-norm for compact, oriented 3-manifolds (with or without a boundary ∂M) was first defined on the second relative homology group H2(M,∂M;Z) of M in Thurston [13]. Any class of this relative second homology group can be represented by a compact, oriented two-dimensional surface S in M. Each such surface has an integer-valued Euler characteristic χ(S) which can be used to define a norm on H2(M,∂M;Z). By Poincaré duality each class of the group H2(M,∂M;Z) determines a class of the first cohomology group H1(M;Z) of M; H2(M,∂M;Z)≅H1(M;Z). Due to this equivalence, the T-norm can also be defined as a norm on H1(M;Z). We use the formulation of the T-norm as a norm on H1(M;Z). In Dunfield [2], this version of the T-norm is defined as follows:

For a compact, connected surface S, let χ−(S)=|χ(S)| if χ(S)≤0 and 0 otherwise. For a surface with multiple connected components S1,S2,…,Sn let χ−(S) be the sum of the χ(Si). Then the T-norm of an integer-valued class ϕ∈H1(M;Z)≅H2(M,∂M;Z) is

||ϕ||T={infχ−(S)∣S is a %
properly embedded orientedsurface that is dual to ϕ}.

Remark 2.6.

The surface S of this definition may or may not be connected. It is shown in [13] that if cohomology class ~ϕ satisfies ~ϕ=d⋅ϕ for some integer d≥1, then ∥~ϕ∥T=d⋅∥ϕ∥T because ~ϕ represents d disjoint surfaces each with T-norm ∥ϕ∥T. The T-norm is additive for disjoint surfaces.

Also in [13], the T-norm is extended using convexity from integer-valued classes ϕ∈(Zb1)∗ to real-valued classes ϕ∈(Rb1)∗ to determine a convex set in (Rb1)∗ called the T-norm unit ball BT. Thus we consider the T-norm as a semi-norm on H1(M;R), the first cohomology group of M with real coefficients, rather than H1(M;Z).

3.1. Link splicing

We use the notation that L=(Σ,L) denotes the link L and its ambient space Σ which is a homology 3-sphere. Given two links, L in Σ and L′ in Σ′, with r and s components respectively we may form the link L′′ in Σ′′ with r+s−1 components by selecting link components, S of L and S′ of L′ and splicing the two links together along S and S′. To do this, first assume that S and S′ have tubular neighborhoods with meridian and longitude (m,l) and (m′,l′) respectively. To splice L to L′ along S and S′ we attach m to l′ and l to m′.

m→l′,

l→m′.

The splice is a homeomorphism of a tubular neighborhood of S which is a solid torus, D2×S1, to a tubular neighborhood of S′, also a solid torus, S1×D2, along a boundary torus S1×S1. The action can be represented as D2×S1∪S1×S1S1×D2. This union of two solid tori across a torus induces the appropriate map in homology connecting each meridian with a longitude.

3.2. Seifert links

Seifert-fibered spaces

To construct a Seifert-fibered space, we start with a solid torus T and remove n parallel tori to obtain the space Tn. The fundamental group of this space Tn is given by

π1Tn=⟨h,y0,y1,…,yn∣[h,yi]=1,(0≤i≤n),y0⋅y1⋯yn=1⟩.

This is an S1-fibration of Tn with base space a disc with n points removed and circle fibers. h and y0 represent the longitude and meridian of the ambient solid torus T. Each yi,i≠0, represents a meridian of ith torus determined by the ith hole of the disc. The commutation relations say that each generator yi,i=0,…,n commutes with the longitude h of T which means the fibration has trivial monodromy. Next we attach n+1 solid tori, Vi, to Tn along the boundary tori in a special way using attaching maps that are homeomorphisms.

M(e;(α1,β1),…,(αn,βn))=Tn∪n⋃i=0Vi.

Each meridian generator, mi, of the solid torus, Vi, is mapped to αiyi+βih for i≠0. For the solid torus V0 the meridian generator, m0 is mapped to y0+eh where e is an integer:

mi=αiyi+βih,i=1,…,n.

m0=s0+eh.

The space we obtain by this procedure is called the Seifert-fibered space
M(e;(α1,β1),…,(αn,βn)). It has n exceptional fibers of type (αi,βi),i=1,…,n) where αi and βi denote the number of times the ith exceptional fiber wraps around the ith torus longitudinally and meridianally respectively. The exceptional fibers are the core circles of the solid tori after we have attached them; if Vi=D2×S1, then the core circle of Vi which becomes the ith exceptional fiber is 0×S1.
This space has fundamental group with presentation

π1(M(e;(α1,β1),…,(αn,βn)))

=

⟨h,y1,…,yn∣yαiihβi=1,[h,yi,]=1,

(0≤i≤n),y1⋯ynh−e=1⟩.

It is equivalent to the fundamental group of Tn with the one additional relation added for each of the solid tori Vi,i=0,…n attached. This construction of Seifert-fibered spaces is taken directly from Zieschang [17].

Seifert link:

We obtain an r-component Siefert link from the Seifert-fibered space M(e;(α1,β1),…,(αn,βn)) by removing a tubular neighborhood from each of the first r exceptional fibers. Each component Si, with i=1,…,r, has a complement in the ambient Seifert-fibered space which is equivalent to a torus knot, labeled by (αi,βi). We leave the remaining n−r components alone so that the complement of link has n−r singular fibers. This link is denoted L=S1∪⋯∪Sr. The ambient Seifert-fibered space and L together form the pair L=(M(e;(α1,β),…,(αn,βn)),S1∪⋯∪Sr). Thus L denotes the link and L denotes both the ambient space Σ=(M(e;(α1,β),…,(αn,βn)) the link lies in and the link L itself; L=(Σ,L). The following is an equivalent but more concise definition of a Seifert link.

Let L be a link in a 3-manifold M and let the interior of a closed tubular neighborhood N(L) of L be denoted intN(L). Then L=(M,L) is a Seifert link if the link exterior, which is M∖intN(L), of L in M is a Seifert-fibered space.

Remark 3.4.

If the Seifert link L is in the 3-sphere S3, we drop the bold-faced notation so that L=(S3,L):=L in this case.

3.5. Seifert-fibered homology 3-spheres

If we require that the Seifert-fibered space has the homology of the 3-sphere we must have the additional relations that

n∑i=1βiα1…^αi…αn=1 %
and e=0.

The notation ^αi means to remove the component αi from the equation. In this case it is not necessary to include the coefficients βi,i=1,…,n in uniquely labeling the Seifert-fibered space so that if the space is in a homology 3-sphere we use the notation Σ(α1,…,αn) and call it an unoriented Seifert-fibered homology 3-sphere of type (α1,…,αn). We use the notation Σ(ϵ;α1,…,αn) with ϵ=±1 to indicate the two possible orientations of the Seifert-fibered homology 3-sphere. Thus we use oriented Seifert-fibered homology 3-spheres. The ambient oriented Seifert-fibered homology 3-sphere together with link L=S1∪⋯∪Sr form the pair denoted L=(Σ(ϵ;α1,…,αn,),S1∪⋯∪Sr). This is the notation we shall use for all Seifert links which we assume are in oriented Seifert-fibered homology 3-spheres. The Seifert link L as defined above is an r-component link with first homology group given by Zr as would be the case if it was instead in the 3-sphere. To a given unoriented Seifert-fibered homology sphere, there exists a unique unordered n-tuple of coprime integers (α1,…,αn) with αi≥2,∀i.

3.6. Graphs of Seifert links and splice diagrams

We can represent a Seifert link in a Seifert-fibered homology 3-sphere as a graph with the following components.

Boundary vertices: \-−−−−∙

The boundary vertex corresponds to the solid torus which is a neighborhood of an exceptional fiber labeled by αi.

Arrowhead vertices: \-−−−⟶

The arrowhead vertex corresponds to one of the link components where a tubular neighborhood of an exceptional fiber has been removed. It also has the label αi.

Nodes: ⊕ and ⊖

A node corresponds to a Seifert manifold embedded in a link exterior; the edges incident to a node correspond to the boundary components of the Seifert manifold, or for edges that lead to boundary vertices or arrowhead vertices to boundaries of tubular neighborhoods of fibers. Within each node we insert + and − corresponding to the two possible orientations of the Seifert manifold. Each node must have at least three edges incident on it.

We can now form a splice diagram (graph) by connecting two arrowhead vertices of the graphs of a pair of Seifert-fibered links. An example of a splice diagram is shown in Figure 1.
A splice diagram determines an E-N graph link which is defined as follows:

Definition 3.7.

An E-N graph link consists of either a Seifert link in a homology 3-sphere or the splice of two or more Seifert links in homology 3-spheres.

In the rest of this article, we assume that the graph of graph link L has r arrowhead vertices labeled v1,…,vr, p nodes labeled vr+1,…,vr+p and q boundary vertices labeled vr+p+1,…,vr+p+q. We also assume that the graph link is irreducible; it can not be expressed as a disjoint sum. In addition, by the term graph link we mean an E-N graph link.

3.8. Alexander polynomial of a graph link

The Alexander polynomial of the graph link L is directly determined by its graph Γ.

Assume Γ (the graph of L) is connected. Then the Alexander polynomial of the graph link L is

(3)

△L(t1,…,tr)=r+p+q∏j=r+1(tl1j1tl2j2…tlrjr−1)δj−2.

If r=1, so that the link is a knot, the formula is

(4)

△L(t1)=(t1−1)p+q+1∏j=2(tl1j1−1)δj−2.

Any terms of the form (t01…t0r−1)d (d denotes an integer) which may occur on the right-hand side of these two equations should be formally canceled against each other before being set equal to zero.

The Alexander polynomial has the same number of variables as arrowhead vertices and the product is over all vertices which are not arrowhead vertices; the nodes and boundary vertices. δj indicates the number of edges incident of each vertex.
The linking numbers lij can be obtained directly from the graph. For any two distinct vertices vi and vj of a graph Γ, let σij be the simple path in Γ joining vi to vj, including vi and vj. Then we have that

(5)

lij={(product of all signs of nodes on σij)⋅(product of all edge weightsadjacent to these nodes but not on σij).}

3.10. T-norm of graph a link

The T-norm of the graph link L is also directly determined by the graph of L.

The T-norm of ϕ for the irreducible graph link L, which is not the unknot in S3, is

(6)

∥ϕ∥T=r+p+q∑i=r+1(δi−2)∣∣.r∑j=1ϕjlji∣∣

Remark 3.12.

For a graph link with r components, let d be the greatest common divisor of the r components of the vector ϕ∈Zr. Up to homeomorphism, the surface S that the cohomology class ϕ represents is the disjoint union of d identical, connected, compact, oriented, two-dimensional surfaces each having genus g and r holes. The genus g of each of these surfaces can be determined using the well-known formula for the Euler characteristic of such a surface,

We now prove the main result of this article relating the Alexander and Thurston norms for graph links in homology 3-spheres.

Theorem 4.1.

The Alexander and Thurston norms coincide for all irreducible graph links with two or more components.

Remark 4.2.

By Corollary 8.3 of Eisenbud and Neumann [3], there is a unique minimal splice diagram for each graph link such that every edge weight is non-negative. Hence in this proof without loss of generality we can assume that the edge weights of all the arrowhead and boundary vertices are non-negative; αi≥0,i=1,…,r and i=r+p+1,…,r+p+q. Even more, since a boundary vertex with an edge weight equal to one represents a non-singular fiber, which is not an exceptional Seifert fiber, we can assume that αi≥2 for boundary vertices.

We prove the theorem by directly calculating the A-norm of the graph link L using Equation (3) for the Alexander polynomial and comparing the result to Equation (6) for the T-norm as given in [3]. We derive three new results which are used in the proof. Each of these results involves a sum over the p splice components of the graph of L;

a T-norm decomposition formula,

the Newton polyhedron of the Alexander polynomial of L as a Minkowski sum, and

an A-norm decomposition formula.

4.3. T-norm decomposition formula

To show that the T-norm is a sum of the T-norms of each node of the graph of a graph link we need the following lemma.

Lemma 4.4.

The linking numbers lji of the arrowhead vertices into a node and into a boundary vertex attached to the same node of a graph link differ only by a factor of αi, where αi is the weight of the edge in the graph connecting the boundary vertex indexed by i to the node.

Proof.

The proof follows directly from the formula for the linking numbers given by Equation (5). It says that to find the linking number between an arrowhead vertex and either a node or a boundary vertex we follow the path on the graph connecting the arrowhead vertex to the node or the boundary vertex. Along the way we multiply the product of all the signs of the nodes and also multiply all the edge weights of edges going into each node but not on the path. Since a boundary vertex is attached to the node by hypothesis, it is clear that the paths between any arrowhead vertex and either the boundary vertex or the node it is attached to are the same except the path to the boundary vertex contains the edge connecting that vertex to the node. Hence the weight of that edge, since it lies on the path, does not appear in the linking number between the arrowhead vertex and boundary vertex but it does in the linking number of that arrowhead vertex with the node. Hence the two linking numbers differ exactly by the factor αi which is the weight attached to the edge of boundary vertex vi indexed by i. ∎

This lemma implies the following Corollary 4.5 to Theorem 3.11 which gives a T-norm decomposition formula for the irreducible graph link L:

Corollary 4.5.

The T-norm of graph link L, can be expressed as a sum of the T-norms of the p splice components of the graph of L.

∥ϕ∥T

=

r+p∑i=r+1(~δi−2)∣∣r∑j=1ϕjlji∣∣

=

p∑i=1∥ϕ∥iT,

where ~δi−2=δi−2−∑qik=11αik>0 and
∥ϕ∥iT is the contribution to the T-norm ∥ϕ∥T of the link from the ith splice component of the graph of the link.

Remark 4.6.

We use the notation that qi denotes number of boundary vertices attached to the ith node, so that the boundary vertices vik attached to the ith node can be indexed by k=1,…,qi. In addition, we denote by αik the edge weight of the kth boundary vertex attached to the ith node. In [3], the boundary vertices and the edge weights are ordered in a manner that does not specify which node the boundary vertex is attached to. Hence the notation αi for an edge weight is used in [3], which we have refined to αik.

Proof.

The expression for the T-norm of this corollary can be obtained from the equation of the T-norm given in [3], Equation (6), by a direct application of the Lemma 4.4 relating the linking numbers between arrowhead vertices to a node and the boundary vertices attached to the node. In effect, the terms δi of the original expression are replaced by the terms ~δi and the sum over both nodes and boundary vertices is replaced by a sum over nodes only. To be precise we shall present this proof in detail.

Without loss of generality, we can prove the formula for the T-norm by showing it is true for the (r+1)th vertex which is also the first node that has by hypothesis q1 boundary vertices attached to it. The equation given in [3] for the T-norm of an irreducible graph link, Equation (6), can be written

∥ϕ∥T=r+p∑i=r+1(δi−2)∣∣r∑j=1ϕjlji∣∣−r+p+q∑i=r+p+1∣∣r∑j=1ϕjlji∣∣.

We’ve split the sum into contributions from the nodes first and then the boundary vertices. The contribution to the Thurston norm from the first node is

Applying the Lemma 4.4 that relates the linking numbers of the boundary vertices to the node we have that

lji=1αr+1ilj(r+1),

for j=1,…,r and i=r+p+1,…,r+p+q1.

We can combine the contribution from the node with the contributions from its boundary vertices to obtain

∥ϕ∥1T

=

(δr+1−2−r+p+q1∑k=r+p+11αr+1k)∣∣r∑j=1ϕjlj(r+1)∣∣

=

(~δr+1−2)∣∣r∑j=1ϕjlj(r+1)∣∣.

By repeating this procedure indicated for the (r+1)th node on all p of the nodes we obtain the Equation (4.5) as claimed.

In order to show that ~δi−2>0,∀i, we proceed by induction on the number of boundary vertices attached to the node. Without loss of generality we can show that this relation is true for the first node in order to prove it is true for all p nodes. First, we use that δi≥3 in the definition of a node given in [3] and proceed by induction. Hence consider the first node indexed as i=r+1 and assume that it has only a single boundary vertex attached to it so that q1=1. We have that δr+1≥3 implies that

~δr+1−2=δr+1−2−1αr+11≥1−1αr+11>0.

Next let us assume that ~δr+1−2>0 for q1=n and we will show that this implies ~δr+1−2>0 for q1=n+1. Adding the (n+1)th boundary vertex to the (r+1)th vertex, which is the first node, increases δr+1 by one because of the additional edge into the node and also adds the term (αr+1n+1)−1 to ~δi. Adding this additional boundary vertex adds the strictly positive term αr+1n+1−1αr+1n+1>0 to ~δr+1. By the inductive hypothesis ~δr+1−2>0 without this additional boundary vertex. We find that we must have ~δr+1−2>0 after the addition of the strictly positive contribution of the (n+1)th boundary vertex. Hence by induction ~δr+1≥0 for all values of q1.∎

4.7. Newton polyhedron of the Alexander polynomial of a graph link

We also have to use another application of Lemma 4.4 in order to prove our fundamental theorem. It involves obtaining an expression for the Newton polyhedron N(△L) of the graph link L.

The Newton polyhedronN(f) of a polynomial f is the convex hull of the exponents of the Alexander polynomial △.1 Two Newton polyhedra can be added together using Minkowski addition: The Minkowski sum of two polyhedra K and L is the set of all vector sums x+y with x∈K and y∈L.

The following proposition and corollary state two well-known properties of the Minkowski sum. The second corollary, although somewhat trivial, is new and will be essential in our proof of Theorem 4.11.

Proposition 4.8.

(Gelfand, Kapranov and Zelevinsky [4], Prop. 6.1.2(b)) The Newton polyhedron N(f⋅f′) of the product of two polynomials, f and f′, is given by

N(f⋅f′)=N(f)+N(f′).

Applying this proposition inductively to the product of f with itself we obtain the following corollary.

Corollary 4.9.

The Newton polyhedron N(fn) of a polynomial f to a power n∈N, fn, is given by

N(fn)=n⋅N(f).

Corollary 4.10.

Assume that g divides f for the rational polynomial f/g. Then the Newton polyhedron N(f/g) of this rational polynomial satisfies the relation

(8)

N(f)=N(f/g)+N(g).

Proof.

We use that f can be written as a product of polynomials so that f=fg⋅g and apply Proposition 4.8. ∎

Theorem 4.11.

The Newton polyhedron N(△L(t1,…,tr)) of the graph link L, is given by

N(△L(t1,…,tr))

=

r+p∑i=r+1(δi−2−qi∑j=1αij)N(tl1i1…tlrir−1)

=

r+p∑i=r+1(~δi−2)N(tl1i1…tlrir−1).

Proof.

By Corollary 4.10, given polynomials f and g such that g divides f, their Newton polyhedra satisfy the equation N(f)=N(f/g)+N(g). If we set f=∏r+pi=r+1(tl1i1…tlrir−1)δi−2 and g=∏r+p+qi=r+p+1(tl1i1…tlrir−1), then f/g is the Alexander polynomial △L of L as given in Equation (3). We can prove this corollary by showing directly that N(f/g)=∑r+pi=r+1(δi−2−∑qij=1αij)N(tl1i1…tlrir−1) is a solution of the equation N(f)=N(f/g)+N(g).

In this equation, we’ve also used that N((tl1i1…tlrir−1)δi−2)=(δi−2)N(tl1i1…tlrir−1) since N(fn)=nN(f),∀n∈N, by Corollary 4.9. By Lemma 4.4, the linking numbers of the boundary vertices are the same as the node they are attached to up to a factor of (αij)−1. Using this result in N(g), we obtain

The Newton polyhedra of this sum are all line segments with endpoints (l1iαij,…,lriαij) and (0,…,0). It is geometrically clear that αij is a scaling factor of each of these segments which reduces the length of the segment but does not change its direction. Hence we have the equality

N(tl1iαij1…tlriαijr−1)=1αijN(tl1i1…tlrir−1).

We now substitute this expression for N(g) along with our candidate solution for N(f/g) into Equation (10) and obtain

r+p∑i=r+1(δi−2)N(tl1i1…tlrir)

=

r+p∑i=r+1(δi−2−qi∑j=11αij)N(tl1i1…tlrir−1)

+r+p∑i=r+1qi∑j=11αijN(tl1i1…tlrir−1)

=

r+p∑i=r+1(λiN(tl1i1…tlrir−1)+λ′iN(tl1i1…tlrir−1)).

In this equation we’ve introduced the constants λi=δi−2−∑qij=11αij and λ′i=∑qij=11αij each of which multiplies the same line segment N(tl1i1…tlrir−1). The two terms on the right of this equation can be combined using Minkowski addition provided that λi and λ′i are non-negative for all i. By Corollary 4.5, λi>0,∀i. As mentioned in Remark 4.2, we may assume, without loss of generality, that edge weights for boundary vertices satisfy the inequality αij≥2,∀i,j. This implies that λ′i>0,∀i. Hence since all the constants λi and λ′i which multiply the same line segment are positive, we can use Minkowski addition to combine the two expressions on the right of this equation and the sums involving αij cancel each other.

A polyhedron which is the Minkowski sum of line segments is called a zonotope.

Corollary 4.12.

The Newton polyhedron N(△L(t1,…,tr)) of the graph link L is a zonotope consisting of p line segments, one for each splice component ot the graph of L.

Proof.

By Equation (4.11), N(△L(t1,…,tr)) is a Minkowski sum of the p Newton polyhedra, (δi−2)N(tl1i1…tlrir−1),i=1,…p. The ith component of this sum, which is the Newton polyhedron of the ith splice component of the graph of L, has only two vertices, the endpoints (δi−2)(l1i,…,lri) and (0,…,0). The convex hull of these two vertices is a line segment. ∎

Remark 4.13.

If the graph link L has only one node, it is a Seifert link. By Corollary 4.12 for N(△L) with p=1, the Newton polyhedron of a Seifert link is a zonotope consisting of a single line segment. Dimca, Papadima and Suciu have found the same result in [1] for the Newton polyhedron of a Seifert link by finding a coordinate system for which the Alexander polynomial is a function of only one variable, which they call its essential variable.

4.14. A-norm decomposition formula

Since the Alexander norm is determined by the vertices of the Newton polyhedron N(△), it can be viewed not only as a norm determined by an Alexander polynomial △, but also as a norm determined by the Newton polyhedron N(△) of the Alexander polynomial △. This means that the notations

∥⋅∥A:

=

∥⋅∥△

=

∥⋅∥N(△)

can also be useful. In Long [6], we determine that the A-norm for the Newton polyhedron N(△) is equal to the width function w of N(△) so that ∥⋅∥N(△)=w(N(△),⋅). In the theory of polyhedra, the width function w is a well-known function that can be proved to be Minkowski linear. This means that for arbitrarily chosen polyhedra P and Q and all λ∈R+, it satisfies following:

Minkowski additivity : w(P+Q,ϕ)=w(P,ϕ)+w(Q,ϕ).

Minkowski scaling: w(λP),ϕ)=λw(P,ϕ).

A proof of the Minkowski linearity of the width function can be found in Long [5].

The equivalence of the width function of N(△) and the A-norm for N(△) along with the Minkowski linearity of the width function imply that the A-norm has the following properties:

∥ϕ∥(N(f)+N(f′))=∥ϕ∥N(f)+∥ϕ∥N(f′).

∥ϕ∥λN(f)=λ∥ϕ∥N(f),∀λ∈R+.

By inductively applying Minkowski linearity, we obtain the following A-norm decomposition formula.

Proposition 4.15.

Assume that the Newton polyhedron N(△) of the Alexander polynomial △ can be written as a Minkowski sum of k component Newton polyhedra, λ1N(f1),…,λkN(fk), with λi∈R+,∀i, so that

N(△)=k∑i=1λiN(fi).

Then the A-norm for N(△) is the sum of the A-norms of each component Newton polyhedron:

(11)

∥ϕ∥A=∥ϕ∥N(△)=k∑i=1λi∥ϕ∥N(fi).

Remark 4.16.

In Long [6], we derive a decomposition formula of the Alexander norm ∥⋅∥A=∥⋅∥△ for the polynomial △ expressed as a sum involving the Alexander norms for each the irreducible factors of △; ∥ϕ∥△=∑ki=1ni∥ϕ∥fi for △=fn11…fnkk and ni∈N,∀i.

By substituting the expression for the Newton polyhedron N(△L) of L, Equation (4.11), into the A-norm decomposition formula, Equation (11), we obtain as a corollary the A-norm decomposition formula for graph links.

Corollary 4.17.

The A-norm of graph link L, can be expressed as a sum of the A-norms of the p splice components of the graph of L.

∥ϕ∥A

=

∥ϕ∥N(△L(t1,…,tr))

=

r+p∑i=r+1(~δi−2)∥ϕ∥N(tl1i1…tlrir−1)

=

p∑i=1∥ϕ∥iA

where ∥ϕ∥iA denotes the contribution to the Alexander norm from the ith splice component of the graph of L.

4.18. Proof of the main theorem

Proof.

We prove the theorem by showing that Equation (4.17), the A-norm decomposition formula for graph links, is equal to Equation (4.5), the T-norm decomposition formula.
Each term in the sum of the A-norm decomposition formula for graph links involves the A-norm of a Newton polyhedron N(tl1i1…tlrir−1) which is a line segment with endpoints at the origin and at (l1i,…,lri). The A-norm for the line segment N(tl1i1…tlrir−1) can be obtained by direct application of the Definition 2.2 of the A-norm for the polynomial tl1i1…tlrir−1. This polynomial has two exponents α=(l1i,…,lri) and β=(0,…,0). The supremum which occurs in this definition must take its value on the difference of these two exponents since this polynomial has only two terms. Hence, we have that